Eigenvalues & Eigenvectors

An eigenvector of a square matrix a is a non-zero vector v whose direction a leaves unchanged: a * v = λ * v for some scalar λ, its corresponding eigenvalue. Geometrically, a only stretches or shrinks v (by a factor of λ) rather than rotating it into a different direction. Eigenvalues show up throughout numerical linear algebra: a matrix’s condition number (see Condition Number) is the ratio of its largest to smallest singular value — themselves the square roots of the eigenvalues of aᵗ * a — and stability analysis of iterative systems generally comes down to whether relevant eigenvalues stay inside or outside the unit circle.

rustebra doesn’t compute eigenvalues via a direct, closed-form decomposition (the way it computes LU or QR). Instead, Krylov Methods provides two iterative methods: power_iteration, which converges to the dominant (largest-magnitude) eigenvalue and a corresponding eigenvector, and inverse_power_iteration, which converges to whichever eigenvalue lies nearest a caller-chosen shift — useful for targeting the smallest eigenvalue, or any eigenvalue you already have a rough estimate for. See that section for the algorithms, their convergence behavior, and worked examples.

Gotchas

• These are iterative approximations, not exact decompositions — they return an error (MaxIterationsExceeded) rather than a result if convergence criteria aren’t met within the given iteration budget, and convergence speed depends on how well-separated the target eigenvalue is from its neighbors. See Power Iteration for the specifics.